Normal Distribution Question(s)

[Hooligan]

Suppose the weight of oranges from an orange grove has population mean 230 g and population standard deviation 9 g. Assume that the weight follows very approximately a normal distribution.
μ=230g
σ=9g
(a) What proportion of oranges will be less than 220 g?
z = (x- μ)/σ
z = (220-230)/9
z = -1.11
So,Pr⁡(z< -1.11)=0.1335
∴The proportion of oranges that will be less than 220g is approximately 13.35%

(b) Suppose that a sample of 16 oranges is selected. What is the probability that the sample mean will be at least 235 g?

I can't seem to get the second part of the question...  Maybe I've over thought it? Thanks for your help in advance guys.

[mikee65]

OK Ive gone over a range of ways to approach this and im still not sure if im right, at the beginning I was omitting the fact it was a sample of sixteen, then I started looking at that aspect and yeah. So I really still am not sure, if you have the answer that would be great.

Ok so Pr(X>235) is 0.289, lets make that p so our success, because we want the Pr of the mean is 235, that means that .5 must lie more that 235 and .5 less than, so we do (16 choose *(.289)^8*(.711)^8. which gives us 0.040442. Yea, thats how i see it, but im probably wrong. Ask true tears, he'll answer it immediately ahah. Yeah just looked over it again and i know im wrong because it says at least 235. Yea, im stumped.

[vea]

(from CAS)

Let M be the no. of oranges that have weights over 235g.



(from CAS)

EDIT: I pretty much interpreted it the same way mike did except I fixed it up a bit, someone confirm please.

[sajib_mostofa]

Why's M>8 ?

[vea]

Why's M>8 ?

M greater than or equal to 8 means that there are 8 or more oranges that have weights over 235g.
i.e. The mean of the sample is over 235g.

[sb3700]

Why's M>8 ?

M greater than or equal to 8 means that there are 8 or more oranges that have weights over 235g.
i.e. The mean of the sample is over 235g.

Not true, that means the median is over 235g. If I have 15 with a weight on 234.9g and 1 with a weight of 900g, my mean is still above 235g.

[vea]

Why's M>8 ?

M greater than or equal to 8 means that there are 8 or more oranges that have weights over 235g.
i.e. The mean of the sample is over 235g.

Not true, that means the median is over 235g. If I have 15 with a weight on 234.9g and 1 with a weight of 900g, my mean is still above 235g.

Thanks for clearing that up, what needs to be done in my working out to make it correct?

[vea]

Ah this is such an annoying questing...

[mikee65]

yea its making me anxious i have a CRVs sac tmos, have to ask my teacher this q tmos definitely, ill post up the answer.

[sb3700]

I think what vea did is what the questioner is looking for. When I did Methods, there were no questions like what the question as given is actually asking.

It's asking you to sample a normal distribution. Let Y be the sample mean of the 16 oranges.

I think Y is still normally distributed with a mean of 230 g still (intuitively, aka I'm guessing). The standard deviation would probably be different and would be something like (population st dev) / (degrees of freedom)

Degrees of freedom = (sample size) - 1 = 15, so the st dev of Y would be 9/15 g. Basically, it means if you sample 10000 oranges instead of 16, your st dev of the sample mean will be smaller (which makes sense if you think about it).

So then Y~N(230, 9/15) and you can work it out.

I haven't learnt this kind of statistics properly, and when I did learn it tangentially, it was over a year ago so all of this could be completely wrong btw. All of this is not Methods stuff. If you see this question, assume its actually asking for the median and do what vea did.

[Hooligan]

Thanks soo much for your help guys.

I look forward to what your teacher says Mike!!

Oh, and this question came from a task our teacher gave us, so I dont have the answers...